# Analysis of atmospheric flow over a surface protrusion using the turbulence kinetic energy equation

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## Abstract

Flow over surface obstructions can produce significantly large wind shears such that adverse flying conditions can occur for aeronautical systems (helicopters, STOL vehicles, etc.) Atmospheric flow fields resulting from a semi-elliptical surface obstruction in an otherwise horizontally homogeneous statistically stationary flow are modelled with the boundary-layer / Boussinesq-approximation of the governing equation of fluid mechanics. The turbulence kinetic energy equation is used to determine the dissipative effects of turbulent shear on the mean flow. Mean-flow results are compared with those given in a previous paper where the same problem was attacked using a Prandtl mixing-length hypothesis. The diffusion and convection of turbulence kinetic energy not accounted for in the Prandtl mixing-length concept cause departures of the mean wind profiles from those previously computed, primarily in the regions of strong pressure gradients.

Iso-lines of turbulence kinetic energy and turbulence intensity are plotted in the plane of the flow. They highlight regions of high turbulence intensity in the stagnation zone and sharp gradients in intensity along the transition from adverse to favourable pressure gradient.

## Keywords

Turbulence Kinetic Energy Turbulence Intensity Wind Shear Wind Profile Turbulent Shear## Nomenclature

*a*_{1}proportionality constant of shear stress to turbulence kinetic energy

*b*height of elliptical body

*E*_{0}reference turbulence kinetic energy

*e*dimensionless turbulence kinetic energy (

*e*^{*}/*E*_{0})*G*function of

*z*^{*}/*δ**k*effective viscosity

*L*mixing length

*l*Prandtl mixing length

*P*dimensionless pressure (

*P*^{*}/_{ϱ}*U*_{∞}^{ϱ})- Sc
_{e} ratio of turbulent eddy viscosity to a diffusion coefficient for energy transfer

*u*dimensionless velocity component in

*x*direction (*u*^{*}/*U*_{∞})*u*_{*}friction velocity,\(\sqrt {\tau /\varrho }\)

*û*turbulence intensity

*U*_{e}dimensionless velocity along streamline (

*U*^{*}_{ e }/*U*_{∞})*U*_{∞}reference velocity

*υ*dimensionless velocity component in

*y*direction (*υ*^{*}/*U*_{∞})*w*dimensionless velocity component in z direction (

*w*^{*}/*U*_{∞})*x*dimensionless horizontal coordinate parallel to direction of flow (

*x*^{*}/*b*)*z*dimensionless vertical coordinate (

*z*^{*}/*b*)*α*aspect ratio of ellipse

*γ*intermittency

*δ*boundary-layer thickness

*ε*dissipation term of turbulence kinetic energy equation

*κ*von Kármán's constant

*λ*length of separation bubble

*υ*kinematic viscosity

*ϱ*mean flow density

*σ*_{u}root-mean-square value of fluctuating

*u*velocity component*σ*_{V}root-mean-square value of fluctuating

*υ*velocity component*σ*_{w}root-mean-square value of fluctuating

*w*velocity component*Τ*turbulent shear stress

*υ*convection velocity

## Superscripts

***dimensional quantity

- ′
turbulent fluctuating component

- ¯()
ensemble average operator

## Subscripts

- 0
initial value from undisturbed upstream flow

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## References

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